What is the radius of a solid sphere that has the same mass as a solid cylinder with a radius of 40 cm?

To determine the radius of a solid sphere that has the same mass as a solid cylinder with a radius of 40 cm, we need to use the formulas for the volume of both shapes and the relationship between mass, density, and volume.

First, let’s calculate the volume of the solid cylinder. The volume (V_cylinder) of a solid cylinder is given by the formula:

Vcylinder = πr2h

where r is the radius and h is the height of the cylinder. Assuming the height is h cm, the volume of the cylinder becomes:

Vcylinder = π(40 cm)2h = 1600πh cm3

Next, we need to find the volume of the solid sphere (V_sphere), which is given by the formula:

Vsphere = (4/3)πr3

where r is the radius of the sphere. To find the mass of both shapes, we can express it in terms of density (ρ):

mass = density × volume

Since we are looking for the case where the mass of the cylinder is equal to the mass of the sphere, we can express this as:

ρcylinder * Vcylinder = ρsphere * Vsphere

If we assume that both the cylinder and the sphere are made of the same material, then their densities will be equal (ρcylinder = ρsphere), which allows us to simplify the equation:

Vcylinder = Vsphere

Now substituting the volumes we calculated earlier gives us:

1600πh = (4/3)πr3

We can divide both sides by π (since it’s a common term) and rearrange to find r:

1600h = (4/3)r3

Next, isolating r3:

r3 = (1600h * 3)/4 = 1200h

Finally, taking the cube root of both sides gives:

r = (1200h)(1/3)

Thus, the radius of the solid sphere is dependent on the height of the cylinder (h). For example, if the height of the cylinder is 1 cm, the radius of the sphere would be:

r = (1200 * 1)(1/3) ≈ 10.6 cm

In conclusion, the radius of the sphere can be found using the height of the cylinder, and you can adjust the height according to your specific scenario to find the corresponding radius of the sphere.